Caldeŕon-Zygmund Operator on Space of Homogeneous Type

In the 1970's, in order to extend the theory of Caldeóon-Zygmund singular integrals to a more general setting, R. Coifman and G. Weiss introduced certain topological measure spaces which are equipped with a metric which is compatible with the given measure in a sense which will be detailed in this chapter. These spaces are called spaces of homogeneous type. In this chapter we present the major notational conventions and basic results of the theory of Calderón-Zygmund operators on spaces of homogeneous type. As we already noticed, it becomes indispensable to have a criterion for L ² continuity, without which the theory collapses like a house built on sandy beach. One such criterion is the T1 theorem of G. David, J. L. Journé and S. Semmes on spaces of homogeneous type. Before proving the T1 theorem of G. David, J. L. Journé and S. Semmes, we will explain the Littlewood-Paley analysis on spaces of homogeneous type, which, based on Coifman's idea on decomposition of the identity operator, was developed by the above authors. The Littlewood -Paley analysis on spaces of homogeneous type becomes a starting point to provide wavelet expansions of functions and distributions. This will be addressed in Chapter 3.